Development of bead-spring polymer models using …web.mit.edu/doylegroup/pubs/UnderhillJoR2005.pdfDevelopment of bead-spring polymer models using the constant extension ensemble Patrick

Development of bead-spring polymer models usingthe constant extension ensemble

Patrick T. Underhill and Patrick S. Doylea)

Department of Chemical Engineering, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139-4307

(Received 23 November 2004; final revision received 24 June 2005�

Synopsis

We have examined a new method for generating coarse-grained models of polymers. The resultingmodels consist of bead-spring chains with the spring force law taken from the force-extensionbehavior in the constant extension ensemble. This method, called the polymer ensembletransformation method, is applied to the freely jointed chain. The resulting model illustrates whycurrent bead-spring chain models are insufficient in describing polymer behavior at highdiscretization. Applying the method to the freely jointed chain with unequal rod lengths showedthe effect of varying flexibility in the chain. The method was also used to generate a bead-springmodel of F-actin, which shows how the method is not restricted to one molecular model andcan even be applied to experimental data. The current limitations of the method are discussed,including the need for approximate bending potentials to model the worm-like chain witha bead-spring chain. We discuss practical issues such as using the bead-spring models in Browniandynamics simulations and develop a simple spring force law that can accurately represent a freelyjointed chain with only a few rods per spring. Because of the functional form of this newforce law, existing computer simulations can be easily modified. © 2005 The Society ofRheology. �DOI: 10.1122/1.2008294�

I. INTRODUCTION

Polymer kinetic theory �Bird et al., 1987� has become an important tool to understandrheology of polymers as well as to understand single polymer molecule experiments�Doyle et al. �2000�, Hur et al. �2000�; Larson et al. �1997��. The most detailed modelstypically used to study flow properties are the freely-jointed chain �FJC� �Flory �1989��and Kratky-Porod �Kratky and Porod �1949�� models. These models assume that thepolymer at smaller length scales remains at local equilibrium. The remaining effect of theatomistic makeup of the polymer is the determination of the Kuhn length or persistencelength. Within this assumption of local equilibrium these detailed models would presum-ably have the desired accuracy but are computationally intractable for the long polymersand long time scales of interest. A reduction of the degrees of freedom is possible if thesemodels are coarse-grained further to bead-spring chains.

Initially few springs were used and the polymers contained many persistence lengthsso the spring force laws were obtained by examining the behavior of the detailed modelsfor very large number of persistence lengths. This led to the derivation of the Marko and

a�
Author to whom all correspondence should be addressed; electronic mail: [email protected]
© 2005 by The Society of Rheology, Inc.963J. Rheol. 49�5�, 963-987 September/October �2005� 0148-6055/2005/49�5�/963/25/$25.00

964 P. T. UNDERHILL AND P. S. DOYLE

Siggia interpolation formula �Marko and Siggia �1995�� and the inverse Langevin forcelaw. Comparison with single molecule experiments have shown that the global behaviorof long molecules is well approximate by these force laws in the so-called entropic region�Bustamante et al. �1994��. Recently, it has become more common to use a larger numberof springs. When modeling a polymer with the same contour length with a larger numberof springs, a spring must represent a smaller segment of polymer. To date, these submol-ecules are still being modeled with the Marko and Siggia or inverse Langevin force law.However, this has been shown to be in error if each submolecule no longer contains alarge enough number of Kuhn lengths or persistence lengths �Underhill and Doyle�2004��.

If each spring is chosen to model only a few Kuhn lengths of a freely jointed chain,the necessary force law can look strange compared to conventional spring force laws.This may seem confusing if the whole polymer contains many Kuhn lengths. Consider,for example, a freely jointed chain with 200 rods. If one examined the average force as afunction of extension for the whole chain, it would be very close to the inverse Langevinfunction. However, if one models this polymer with a bead-spring chain containing 100springs, the spring force law needed for each spring is dramatically different from theinverse Langevin function. Not only is this strange force law needed to model the be-havior of the freely jointed chain on the scale of two rods, but also it is necessary in orderto get the correct overall force-extension behavior and rheological behavior for the wholechain.

The increase in the number of springs is not simply motivated by increasing compu-tation power, but may be necessary to capture essential physics of interest. Increasing thenumber of springs increases the number of relaxation modes of the polymer and distrib-utes the viscous drag more evenly along the contour. In addition, recently a number ofapplications have surfaced with microdevices in which the behavior of the polymer at asmaller length scale is of critical importance �Doyle et al. �2002�; Jendrejack et al.�2003�; Han and Craighead �2000��. Accurately modeling this behavior requires having acoarse-grained model that can reproduce the behavior at that length scale quantitativelyyet also is computationally tractable.

Here we discuss how to systematically determine the spring force law that should beused if each spring does not represent a large number of Kuhn lengths. In this way webridge the gap between detailed models such as the FJC and Kratky-Porod, and bead-spring chains with a large number of persistence lengths per spring. While these modelswill be guaranteed to reproduce some specified property of the polymer, note that if alarge number of beads is used, they may still not be computationally feasible. Our ap-proach will start from a model such as the FJC and determine the spring force law neededfor a bead-spring chain model without assuming each spring represents a large number ofKuhn steps. We will apply this method to a number of “toy problems.” Each of theseexamples illustrates an important point about coarse-graining into bead-spring chains.

We also discuss some practical issues of using these force laws. In some situations itmay be easier to use models that are not accurate in all situations. For example, if longflexible polymers are near equilibrium, Hookean springs are sufficient to capture theresponse. For long DNA, freely jointed chains can capture much of general response overlarge length scales except gives different approaches towards full extension. Other typesof approximations include things like using the FENE or Cohen force laws �Warner,�1972�; Cohen �1991�� as approximations of the inverse-Langevin function or using rigid
rods �constraints� instead of very stiff Fraenkel springs.

965DEVELOPMENT OF BEAD-SPRING POLYMER MODELS

II. METHOD

The general method we will be using was first introduced by Underhill and Doyle�2004�. For completeness we give here a brief review of the method and its justification.An overview of the coarse-graining approach is shown in Fig. 1, comparing it with thecurrent bead-spring chain models. We require that the model behaves identically whenplaced in the same external environment. The behavior we consider here is the force-extension behavior in the constant force ensemble �how much the molecule extends underapplication of a constant force�, �z��f�. The Marko and Siggia and inverse Langevin forcelaws correspond to using directly the constant force behavior of the polymer. Our methodinstead uses the constant extension behavior of the polymer, so we call it the polymerensemble transformation �PET� method.

While one reason the constant force ensemble was used previously to obtain a springforce law is the ability to perform analytic calculations, it may be comforting that thevery behavior trying to be modeled �the constant force response� is used directly inbuilding the bead-spring model. In fact, it may seem counterintuitive to use the constantextension ensemble to build the bead-spring chain considering both the real polymer andthe beads in the model are free to move. The resolution of this paradox is illustratedpictorially in Fig. 2.

We first note that the behavior of the polymer is an average over all possible configu-rations. Consider performing that average by choosing a series of reference points on thepolymer where the beads in the bead-spring model will be �shown by black circles in Fig.2� and separate all the polymer configurations into categories for which each category hasfixed reference points. First the average is performed within each category, for which thereference points are fixed, thus the constant extension ensemble is needed. In this way wereplace each category �which contains many configurations� by a single configuration ofthe bead-spring chain.

We can show mathematically that this method works by examining the partition func-tion of the polymer. If the bead-spring chain produced has the same partition function asthe polymer being modeled, then the model will reproduce the polymer behavior. This is

FIG. 1. There are multiple paths to build a bead-spring model. The goal of coarse-graining is to produce amodel such that its behavior, �z�m�f�, matches the polymer’s behavior, �z�p�f�. The paths �arrows� representdifferent methods of taking a property of the polymer to be the spring force law. The PET method introducedhere, path I, uses the constant extension polymer behavior as the spring force law. The conventional method,path II, uses the constant force behavior as the spring force law.

similar to coarse-graining from quantum chemistry calculations to repeat unit level mod-


els using the distribution function of the detailed model to determine the potential energyof the coarser model �Baschnagel et al. �2000��. The partition function of the polymer inthe constant force ensemble is

Z�f� =� exp�− U + f · Rtot

kBTdV , �1�

where U is the total internal energy, f is the constant external force applied to the end ofthe polymer, Rtot is the total end-to-end vector of the polymer, kB is the Boltzmann’sconstant, T is the absolute temperature, and the integral is taken over all configurationspace of the polymer.

For simplicity, let us first examine the development of a single spring �dumbbell�model. By using a Dirac delta function we can introduce the new variable r and canrewrite the partition function as

Z�f� =� ��r�exp� f · r

kBTdr , �2�

where

��r� =� exp�− U

kBT��r − Rtot�dV �3�

is the constant extension ensemble partition function for which the end-to-end vector isheld constant at r. One can verify this equivalence by inserting Eq. �3� into Eq. �2� anddoing the r integral. The partition function in Eq. �2� is of the same form as the partitionfunction for a single dumbbell, which is given by

Z�f� =� exp�− Us�r�kBT

exp� f · r

kBTdr . �4�

FIG. 2. The physical justification for the PET method is based on sorting all configurations of the polymer intocategories. Within each category the reference points �black circles� are held fixed. The lines signify differentpolymer configurations within the category. When performing the average over the configurations in a category,the segment between reference points is effectively in the constant extension ensemble. This segment is re-placed by a spring in the bead-spring model.

If the spring potential energy in the dumbbell is


Us�r� = − kBT ln ��r� , �5�

then the model will have the same partition function �and thus equilibrium behavior� asthe real polymer. This forms the basis of the PET method, in which the constant extensionensemble behavior of the polymer is used to produce the bead-spring model. While thiswas a derivation for a single dumbbell model, a similar procedure can be used for chains.Each of the spring coordinates can be introduced into the polymer partition functionusing Dirac delta functions. For example, if two springs were desired then after theappropriate transformation the partition function would be

Z�f� =� ��r1,r2�exp� f · �r1 + r2�kBT

dr1dr2, �6�

where ��r1 ,r2� is the partition function of the true polymer given that the vector con-necting the beginning to the midpoint of the contour length is held at r1, and the vectorconnecting the midpoint to the end is held at r2. The partition function for a two springchain is

Z�f� =� exp�− Us�r1,r2�kBT

exp� f · �r1 + r2�kBT

dr1dr2, �7�

where Us�r1 ,r2� is the potential energy of having spring 1 with connector vector r1 andspring 2 with connector vector r2. Note that this energy is not necessarily decoupled intoindependent contributions for each spring. Again we see that if we take

Us�r1,r2� = − kBT ln ��r1,r2� , �8�

then the bead-spring chain model will have the same equilibrium behavior as the truepolymer.

While we have shown that the PET method generates a bead-spring chain that has thesame constant force ensemble partition function as the true polymer, it is not obviouswhether models with more than a single spring will have the same behavior as thepolymer in the constant extension ensemble. In fact, there is a one-to-one correspondencebetween the two partition functions that guarantees that the model will have the correctconstant extension behavior. Writing the partition function for an imaginary force, we seethe integral is a Fourier transform

Z�ikBTk� =� ��r�exp�ik · r�dr . �9�

The constant extension partition function is therefore given by the inverse Fourier trans-form

��r� = 1

2��d� Z�ikBTk�exp�− ik · r�dk , �10�

where d is the dimensionality of the vectors. Thus using the general PET method sum-marized in Eq. �8� reproduces the polymer behavior in both constant force and constantextension ensembles.

Looking at the relation between the constant extension and constant force partitionfunctions, we see that they are related analogously to other conjugate ensembles such asthe microcanonical and canonical ensembles �Pathria �1996��. There has been recentinterest in the constant extension and force ensembles, including what constitutes a “large
system” so the ensembles are equivalent and what concepts from thermodynamics do not


apply for the single molecule analysis �Keller et al. �2003��. While our work here alsodiscusses the two ensembles, it is different from previous discussions. Most previousresearchers studying these ensembles are discussing them in the context of comparisonwith stretching experiments of single molecules. Depending on the constraints imposed inthe experiment, a different ensemble can be appropriate for the analysis. We have shownthat when coarse-graining, the spring force law should be taken from the constant ex-tension ensemble so that the coarse-grained model has the same response as the polymerunder all constraints.

III. RESULTS AND DISCUSSION

A. Freely jointed chain

The first model we discuss is the freely jointed chain in order to review the simplestapplication of the method. The largest difficulty in calculating the spring force needed tomodel a polymer is determining the partition function ��r1 ,r2 ,…�. Because there is agreat simplification in this calculation for the freely jointed chain polymer, we consider ithere. The result, called the random walk spring �RWS� model, is a collection of springforce laws in which the form of the spring force depends on the number of Kuhn lengthsrepresented by each spring.

It is well-known that the probability distribution of a chain with infinitely stiff Fraen-kel springs is the random walk distribution �Bird et al. �1987��. This is the system that wecall here the freely jointed chain. A system with rods as rigid constraints without acorrective metric force would have correlations between rods resulting in a distributiondifferent from the random walk distribution. We do not consider this case here.

The calculation of the partition function, ��r1 ,r2 ,…�, for the FJC is simplified be-cause the joints correspond to free hinges. This decoupling between the segments of thepolymer causes the partition function to be separable into a product of partition functions:

��r1,r2,…� = ��r1��r2� ¯ . �11�

Because the spring potential is chosen from the logarithm of the partition function, thespring potential becomes a sum over each spring �with no cross terms�

Us�r1,r2,…� = Us�r1� + Us�r2� + ¯ . �12�

This is a significant simplification because the problem has been reduced to finding thespring potential energy for only dumbbell models.

The spring force law is calculated by first determining the constant extension partitionfunction given in Eq. �3� or alternatively in Eq. �10�. The result is proportional to thewell-known probability density of a three-dimensional random walk given by Rayleigh’sformula �Rayleigh �1919�� or Treloar’s formula �Treloar �1975��. Using Rayleigh’s inte-gral form, the partition function is

��r� �1

r�

0

�

u sin�ur�� sin�uA�uA

�

du , �13�

where r= �r�, A is the Kuhn length, � is the fully extended length of a spring, and �=� /A is the number of Kuhn lengths represented by each spring. Alternatively, the par-
tition function can be written as Treloar’s summation formula as


��r� �1

r t=0

��− 1�t

t ! �� − t�!�� − �r/A� − 2t

2�−2

, �14�

where the upper limit � is taken from the condition

�� − r/A�/2 − 1 � � �� − r/A�/2. �15�

The spring potential energy to within an arbitrary additive constant is found using Eq. �5�to be

Us�r� = − kBT ln� 1

r�

0

�

u sin�ur�� sin�uA�uA

�

du� . �16�

The spring force is calculated as the derivative of the potential energy where by conven-tion a positive value is a retractive force. Recall that this set of spring force laws exactlyreproduces the behavior of the FJC for integer �, while in current simulations the inverseLangevin function, or some approximation to it, is being used for all �. In Fig. 3 wecompare the force laws from the RWS model with the inverse Langevin function fordifferent values of �.

The first case to consider is �=1. One important property of a coarse-graining methodis that if the level of coarse-graining is reduced to the level of the detailed model, theresult of the “coarse-graining” is the same as the detailed model. The probability of arandom walk after a single step is a delta function, which is reproduced by Eq. �13�. Oneway of getting a spring with a delta function distribution is to use an infinitely stiffFraenkel spring.

We see that for small � the RWS model has special features that are not conventionallyseen in bead-spring chain models. Note that these features are necessary in order for themodel to have the correct behavior. When �=2 the RWS model spring force increaseswith decreasing extension, diverging at zero extension. There is also a discontinuous

FIG. 3. Comparison of spring force laws of the random walk spring model �dashed line� with the inverse

Langevin function �solid line�. The dimensionless axes are fRWS= fRWSA / �kBT� and r=r /�. The results are for�=2 �upper left�, �=3 �upper right�, �=4 �lower left�, and �=8 �lower right�.

divergence at full extension. The functional form is given by


fRWS�r;� = 2� =kBT

r, r � . �17�

For �=3 the force law is zero below one-third extension, at which point there is a jumpdiscontinuity. Above one-third extension the spring force is

fRWS�r;� = 3� =3kBTA

�3A − r�r, �/3 r � . �18�

For �=4 the force has a nonzero limit at zero extension and a discontinuous first deriva-tive at half extension

fRWS�r;� = 4� =3kBT

8A − 3r, 0 r �/2, �19�

fRWS�r;� = 4� =kBT�4A + r�

�4A − r�r, �/2 r � . �20�

The dramatic changes in the form of the force law come about when the true polymertakes configurations of a certain form. They also correspond to changes in the upper limit� in Treloar’s formula, Eq. �15�. To better illustrate this, in Fig. 4 we show � ,Us , fRWS,and sample configurations for �=3. We see that the step discontinuity occurs when theends are separated by a distance equal to one rod. For end-to-end distances less than onerod, the partition function �number of configurations� is constant, resulting in a vanishingforce. Beyond this special point the partition function decreases but with a discontinuousslope. This change in slope causes a jump discontinuity in the force. The change isdiscontinuous because the rods are stiff. If one were to coarse-grain three Fraenkelsprings into a single spring, it would be continuous, with the discontinuity appearing

FIG. 4. Progression from the constant extension partition function to the spring force in the random walk spring

model for �=3. The dimensionless axes are Us=Us / �kBT�, fRWS= fRWSA / �kBT� and r=r /�. Sample configura-tions are shown for the three rod system at fractional extensions of r=1/3 and r=2/3.

when the Fraenkel springs become infinitely stiff.


B. Unequal rod lengths

The freely jointed chain discussed thus far consists of rods of equal length. We canexamine how sensitive the spring potentials are to a distribution of rod lengths, whichwould correspond roughly to a polymer with regions of differing flexibility. If the scaleover which the flexibility changes is larger than the Kuhn length, a freely jointed chainwith changing rod lengths should be a reasonable model. This situation could appear inblock copolymers, atactic polymers, and DNA having blocks with the same repeated basepair. We first must discuss the behavior of the system that is to be modeled. For the equalrod case, the constant force response is the Langevin function for any number of rods.However, the behavior of a general chain is a sum over the response of each rod

�z� = i=1

Nr

�zi� = i=1

Nr

AiL fAi

kBT� , �21�

where i denotes the different rods in the chain which is to be modeled by a spring, Nr isthe number of rods, Ai is the length of rod i, and L is the Langevin function. In general,this summation does not simplify. To understand how this response differs from theresponse with equal lengths, we examine the small and large force limits and try toidentify an effective Kuhn length for which this appears like a single Langevin function.

If we examine the limit when f is large compared to kBT /Ai even for the smallest Ai,the average fractional extension of the chain becomes

�z��

� 1 −kBTNr

f�, �22�

where � is the fully extended length of spring and is equal to the sum of the rod lengths

� = i=1

Nr

Ai. �23�

Comparing this behavior with a system with equal rod lengths gives a high-force effec-tive Kuhn length

Aeff,h =�

Nr=

1

Nr i=1

Nr

Ai = A . �24�

We have introduced the overbar notation for the average over the rods in the chain. If weexpand the fractional extension for small force, the behavior becomes

�z��

�f

3kBT i=1

Nr

Ai2/� . �25�

Therefore the low-force effective Kuhn length is

Aeff,l = i=1

Nr

Ai2/� =

i=1

Nr

Ai2/

i=1

Nr

Ai = A2/A . �26�

The high-force value is simply the average rod length, while the low-force value in Eq.�26� is larger. The larger the spread in rod lengths, the larger the difference between thetwo.

After understanding the constant force response of the bead-rod chain, we can exam-
ine the spring force needed in a bead-spring model that gives the same constant force


response �which itself differs from a single Langevin function�. As shown previously, thisspring force comes from the constant extension response of the bead-rod chain. This canbe written again using Rayleigh’s formula but for a random walk with nonidentical steps.For example, to use a spring to model a three rod system with rod lengths A1 ,A2, and A3

one should use a spring potential energy of

Us�r� = − kBT ln�1

r�

0

�

u sin�ur�sin�uA1�

uA1

sin�uA2�uA2

sin�uA3�uA3

du . �27�

The spring force is the derivative of this potential.Let us examine a specific system to illustrate the behavior of these unequal rod chains.

Figure 5 shows the spring force needed for a series of chains each with m rods of lengthA and m rods of length 5A for m ranging from 1 to 6. Similar to the equal rod case, forshort chains �small m� the spring force laws needed have discontinuities. For example,when m=1 it is impossible for the fractional extension to be less than 2/3. Because apositive force is a retractive force, the force diverges to −� to prevent the fractionalextension from being less than 2/3. Only in the large m limit does the spring force-lawapproach the constant force behavior. Note that even in this limit the spring force neededdiffers from a single Langevin function based on the average rod length.

Figure 5 allowed us to examine how the chains approach the infinitely long limit if therods have a significant difference in lengths. To further examine how different rod lengthsaffect the spring force law needed to model the chain, we will compare chains will thesame contour length but with different ratios of rod lengths. The system contains m rodsof length A and m rods of length pA. The constraint of constant contour length means thatm�1+ p� is held constant. Figure 6 shows an example of the rod system with m�1+ p�=12 and m ranging from 1 to 6. Note that the m=2 case was already seen in Fig. 5, andthe m=6 case has 12 equal length rods. For this example, two effects help make theresponse closer to the inverse Langevin function as m increases; not only does the

FIG. 5. Series of spring forces necessary to model chains with m rods of length A and m rods of length 5A for

m=1, 2, 3, 4, 5, 6, with the arrow denoting increasing m. The dimensionless axes are f s= fs3A /kBT and r=r /�. The dashed line corresponds to m=�. The dotted line is the inverse Langevin function. Note that them=1 force law diverges at r=2/3 and r=1.

number of rods increase with m but also the variance in rod lengths decreases.


In Fig. 6 we scaled the force using the average rod length, which changes with m. Wesaw that for the constant force response this scaling made curves for all distributions ofrod lengths and number of rods have the same high force limiting behavior. Let usexamine closer the constant extension response near full extension. Consider a chain of �rods with lengths A1 ,A2, to A�. We denote the smallest rod length by As. It has beenobserved that in the region �−2Asr� the partition function is

��r� �� − r��−2

r. �28�

This has been explicitly verified for a number of examples for small �, while a generalproof for arbitrary � appears excessively tedious like a related proof given in Appendix Fof Flory �1989�. The spring force needed in this region of extension is

fs�r� = kBT1

r+

� − 2

� − r� . �29�

If we nondimensionalize the force using the average Kuhn length, A, the force as afunction of fractional extension, r, is

fsA

kBT=

1 + �� − 3�r�r�1 − r�

. �30�

Because this force only depends on � and not the exact distribution of rod lengths, the

force is the same in this region as an equal rod system having � rods each of length A.Note that this relation is exact over a finite range which is determined by the smallest rodlength. This can be compared with the constant force response, for which it only matches

the response of � rods each of length A in the limit r→�.By examining the FJC with unequal rod lengths, we have examined a crude model for

chains with variable flexibility. Both the constant force response and the constant exten-

FIG. 6. Series of spring forces necessary to model chains with m rods of length A and m rods of length pA form=1, 2, 3, 4, 5, 6, with the arrow denoting increasing m. The value of p is determined from the condition of

constant contour length, m�1+ p�=12. The dimensionless axes are f s= fs�6/m�A /kBT and r=r /�. The dotted lineis the inverse Langevin function. Note that the m=1 force law diverges at r=5/6 and r=1.

sion response differ from the equal rod case �a single Langevin function�. In other words,


even in the infinite chain limit, the unequal rod system can be distinguished from anequal rod model. Also, throughout the analysis, the order of the rods within a spring didnot matter. This can be seen directly from Eq. �27� for the three rod system. The formulais the same independent of the ordering of the rods.

C. Worm-like chain

The next polymer model we examine is the worm-like chain �WLC�, but initially onlyfor dumbbell models. The next section discusses the extension to multispring chains. Theapplication for a dumbbell follows the general prescription given in Eq. �5�. If the worm-like chain contains a large number of persistence lengths, then the spring force lawreduces to the long chain limit approximated by the Marko-Siggia interpolation formula.However, many biopolymers of interest, such as F-actin and microtubules, do not containmany persistence lengths. In contrast to the FJC, for which exact analytical results can beobtained for short chains, the analytical calculation of the exact distribution function forshort WLCs cannot be done at this time. However, a number of numerical and approxi-mate analytical techniques have been developed recently �Wilhelm and Frey �1996�; Dharand Chaudhuri �2002�; Samuel and Sinha �2002��.

As an example, we calculate the spring force needed to model a single actin filamentusing a dumbbell. Because the end-to-end distance distribution for an actin filament hasbeen measured experimentally �Le Goff et al. �2002��, this example illustrates how theuse of the distribution function to produce a bead-spring model is not restricted to ana-lytical models. The distribution function can come directly from experiments.

Le Goff et al. �2002� show that the distribution for actin matches well to the worm-like chain, and Wilhelm and Frey �1996� develop a good approximation to that behavior.In Fig. 7�a� we show the distribution function of a three-dimensional WLC calculatedfrom the approximate formula given by Wilhelm and Frey �1996� for the actin parametersfound by Le Goff et al. �2002�: L=13.40 m, Ap=16.1 m. In Fig. 7�b� we compare theresulting spring force necessary to reproduce the actin distribution with the Marko andSiggia interpolation formula. The dramatic difference between the force laws illustratesthe importance of choosing the correct one, because the incorrect use of the Marko and

FIG. 7. �a� Constant extension partition function of a single F-actin filament. �b� Comparison of the springforce law determined by the PET method �solid line� for a dumbbell model of an F-actin filament with theMarko-Siggia interpolation formula �dashed line�. The force is made dimensionless using the persistence length,

f s= fsAp /kBT.

Siggia form would result in significant errors.


D. Multispring worm-like chain models

In order to develop a multispring model of the worm-like chain, one must calculate themultidimensional partition function ��r1 ,r2 ,…�. This illustrates a hurdle with imple-menting the PET method. For models such as the worm-like chain with coupling acrossthe reference points �black circles in Fig. 2�, the multidimensional partition function isnot separable, and therefore the spring potential energy is not separable into a sum overeach spring.

However, we can use the same thought process used in Fig. 2 to overcome the diffi-culty in a systematic way. We can eliminate the coupling between the segments by furtherrestricting the category of configurations. The new type of model is illustrated in Fig. 8.In addition to holding points along the polymer with fixed positions, the tangent vector ofthe polymer is held fixed at each of these points. Because the tangent vector is held fixed,there is no coupling across the points. We can replace the category of configurations bya single configuration of a coarse grained model. This new coarse grained model is not atypical type of bead-spring chain, but a series of beads that have a direction associatedwith them �the direction of the polymer’s tangent vector�. The interaction of two neigh-boring beads depends both on the vector connecting the bead centers, r, and the tangentvectors associated with the beads, t1 and t2. The “potential energy” of this interaction is

U�r,t1,t2� = − kBT ln ��r,t1,t2� , �31�

where ��r , t1 , t2� is the partition function in the ensemble where both the separation ofthe ends and tangent vectors at the ends are held fixed.

While this new type of coarse grained model will give the correct equilibrium distri-bution function as the worm-like chain at any level of discretization, it may not becomputationally desirable to perform simulations such as Brownian dynamics �BD� withsuch a model.

We propose that this system can be approximated by a conventional bead-spring chainwith bending potentials. The spring force in this approximation would be taken as theforce in a dumbbell model of the short segment of polymer. This separation is a reason-able first approximation because it correctly models the behavior at the two extremes of

FIG. 8. Restricting the configurations within a category eliminates the coupling in a multispring worm-likechain model. Within each category the reference points �black circles� are held fixed as well as the polymer’stangent vector at the reference point. The lines signify different polymer configurations within the category. Thecoarse-grained model consists of beads which have a vector direction which is the direction of the polymer’stangent vector. The interaction of neighboring beads depends both on the bead positions and the bead directions.

discretization.


In the limit of a large number of persistence lengths between reference points, thecoupling vanishes and thus the bending energy should vanish. In the limit of zero per-sistence lengths between reference points, the polymer acts like a rigid rod. The polymermodel of rigid rods with bending potentials is the Kratky-Porod model if we choose thebending potential to be

Ubend�� =kBTAp

2��2, �32�

where � is the angle between the directions of successive rods, � is the length of the rod,and Ap is the persistence length. In the limit of � much smaller than Ap this approachesthe continuous WLC model. This bending energy could also be used when the springlength, �, is much larger than the persistence length, Ap, because it correctly vanishes inthat limit. We leave a detailed analysis of the decomposition approximation using bend-ing energy and an analysis of how the bending energy varies with �=� /A for future work.

E. Rheology of the models

The force laws presented give the same equilibrium behavior as the polymer includingforce-extension behavior in both the constant force and constant extension ensembles.Therefore it is natural to use these bead-spring chains to model the polymer behavior inother situations such as in flow. We have previously derived the zero-shear properties ofbead-spring chains for arbitrary force law �Underhill and Doyle �2004��. We can usethose results to analyze the rheological properties of these bead-spring chains.

In Underhill and Doyle �2004� it is shown that the retarded motion expansion coeffi-cients, which describe the rheological response in slow and slowly changing flows, canbe written in terms of the force-extension behavior. Note that the analysis is for dilutepolymer solutions and excluded volume and hydrodynamic interactions are not included.There are no bending potentials allowed and the spring force law must be the same foreach spring. The first two retarded motion coefficients, b1 and b2, are

b1 − �s = �0,p = �np�N �LA

12N + 1

N�lim

f→0

�

� f�ztot�m� , �33�

b2 =− �1,0

2= �− np�N �2L2A2

120kBT� 1

3�limf→0

�3

� f3�ztot�m�� N2 + 1��N + 1�

N3 + lim

f→0

�

� f�ztot�m�2� �N + 1��2N2 + 7�

6N2�N − 1� � . �34�

The values of b1 and b2 are related to the viscosity of the Newtonian solvent, �s, thepolymer contribution to the zero-shear viscosity, �0,p, and the zero-shear first normalstress coefficient, −�1,0. The parameters in the response are the number density of poly-mers, np, the number of beads, N, the drag coefficient on a bead, , the polymer contourlength, L, a length proportional to the Kuhn length or persistence length, A, and the ratioof the contour length to A ,�=L /A. The influence of the spring force law enters throughthe derivatives with respect to force of the average extension in the limit of zero force.The average z extension of the model has been made dimensionless using the contourlength L, and the force has been made dimensionless using kBT /A. In general, the de-
rivatives of the force-extension behavior can be a function of �=� /A.


The terms other than the force-extension behavior are contributed to the fact that thehydrodynamic drag is exerted only on the beads and not along the continuous polymercontour. When the number of beads is large enough, these factors disappear. When usingspring force laws such as FENE or Marko-Siggia, the force-extension terms change wheneach spring represents a small segment of polymer. This causes significant errors in therheology which places a limit on how small the springs can be �Underhill and Doyle�2004��. Using the spring laws discussed here eliminates this error. For example, for theRWS model discussed in Sec. III A and choosing A to be the Kuhn length in nondimen-sionalizing the force, the bead-spring chain always has

limf→0

�

� f�ztot�m =

1

3�35�

limf→0

�3

� f3�ztot�m =

− 2

15. �36�

We can also use the retarded motion coefficients to gauge the difference between thedifferent dumbbell models of F-actin shown in Fig. 7�b�. For a dumbbell model, it isuseful to write b1 and b2 directly in terms of the equilibrium averages of the springlength. These moments of the spring length are equal to

�rn�eq =

�0

1

dr rn+2exp�− U�r�/�kBT��

�0

1

dr r2exp�− U�r�/�kBT��, �37�

where U is the spring potential energy. For the dumbbell models, writing b1 and b2 interms of these moments of the spring length gives

b1 − �s = �0,p = np L2

12��r2�eq, �38�

b2 =− �1,0

2= − np 2L4

240kBT��r4�eq. �39�

The force laws in Fig. 7�b� will have different equilibrium averages. The averages for theMarko-Siggia spring can be calculated using numerical integration. Using the Marko-Siggia dumbbell to model an F-actin filament with the contour length and persistencelength from Sec. III C, the averages are �r2�eq=0.41 and �r4�eq=0.21. Alternatively, usingthe dumbbell model that correctly models the worm-like chain distribution has averages�r2�eq=0.77 and �r4�eq=0.61. This means that the spring force laws in Fig. 7�b� have afactor of 2 different �0,p and a factor of 3 different b2.

F. Practical implementation of the models

While the models developed using the constant extension ensemble are advantageousin that they give the correct equilibrium behavior and provide a systematic method forchanging levels of coarse-graining, some practical issues remain for using them in com-
putations such as Brownian dynamics simulations. For small numbers of rods per spring,


the spring force laws can have discontinuities. How do discontinuous force laws performin BD simulations? We address this question by simulating directly using BD the two-rodand three-rod spring force laws at equilibrium.

The two-rod spring force law is proportional to r−1 for allowable extensions. At fullextension, the retractive force must jump to +� to prevent the spring from extending pastthat point. The two features not common to spring force laws are the divergence at smallextension and the discontinuity at full extension. To test the ability to use this force lawin a Brownian dynamics simulation, we use a simple explicit Euler integration scheme�Öttinger �1996��. With this scheme it is possible for the length of the spring at the end ofa time step to be larger than the fully extended length of the spring. This is analogous totwo hard spheres which are not allowed to have a separation distance smaller than theirdiameter. For this spring force law, we implement the same type of algorithm used forhard spheres �Heyes and Melrose �1993�; Foss and Brady �2000��. If the spring is pastfull extension at the end of a time step, we rescale the length of the spring to be at fullextension. This algorithm is known to converge to the correct answer as the time step isreduced. For larger time steps, it is known that this algorithm produces a delta function inthe probability density at full extension with a corresponding depletion at smaller exten-sions.

We tested the algorithm by choosing a random starting configuration and steppingforward in time to equilibrate in a stagnant solvent. We then continued simulating atequilibrium and sampling the magnitude of the extension of the spring. From this webuild a histogram of the probability distribution of the spring length. The spring lengthwas sampled 106 times to reduce the statistical error from finite sampling so most of theerror is due to the nonzero time step. We show in Fig. 9�a� the histograms for differenttime steps compared to the true distribution. The time step is nondimensionalized as

�t � �t kBT

�2 � , �40�

where is the drag coefficient on a bead and � is the fully extended length of the spring.We also show in Fig. 9�b� the convergence of the second and fourth moments of thespring length, �r2�eq and �r4�eq, to the true values. To avoid ambiguity in the sign, we

FIG. 9. This shows the convergence of a BD simulation for a spring force law that models two rods of a freelyjointed chain. �a� Histogram of the probability density of the spring length from a BD simulation compared with

the exact distribution. The time steps used were �t=10−3 ��, �t=10−4 ��, and �t=10−5 ��. �b� The relativeerror in the second moment �r2�eq �� and the fourth moment �r4�eq �� from the BD simulations.

define the relative error as


� �true value − calculated value

true value. �41�

The three-rod spring force law also has a feature not common to force laws. It has a finitejump in the force at 1 /3 extension. This jump is equivalent to a discontinuous slope in theprobability density. We again test the ability of a simple explicit Euler scheme to capturethis feature. Even though the spring force law does continuously diverge at full extension,with this scheme it is possible to end a time step past full extension. While there existmore sophisticated schemes for handling this, we opt for the simple algorithm of rescal-ing the spring to be at 0.99999 of full extension.

We formed the probability density histogram in the same way as for the two-rodspring. In Fig. 10�a� the histograms for different time steps are compared to the truedistribution. We also show in Fig. 10�b� the convergence of the second and fourth mo-ments of the spring length to the true values.

These two examples show that it is possible to simulate the new spring force lawsusing simple BD algorithms. More elaborate schemes may be able to increase the size oftime steps although at the increased computation cost of a more sophisticated algorithm.

The other main practical issue concerns the spring laws when each spring represents alarger number of springs. Although we have shown that the spring force from the RWSmodel �calculated from Eq. �16�� exactly represents the freely jointed chain, and similarlyfor the unequal rod case, the analytical expressions for the spring force laws becomeincreasing complicated as each spring represents more rods. We have seen that the springforce law has different forms in different regions of fractional extension. For the equalrod length case the number of different regions is 1 /2 of the number of rods per spring�rounding up to the nearest integer�. Thus as the number of rods per spring increases, thenumber of different regions increases dramatically. Within each of these regions, theforce law is a rational function where the degree of the polynomials typically increasesrapidly as the number of rods per spring increases. The large number of regions and thecomplexity within each region decreases the practicality of these force laws for largenumber of rods per spring. However, in this limit the spring force laws are becomingsmooth, making it likely that approximate force laws can be developed in the same spirit

FIG. 10. This shows the convergence of a BD simulation for a spring force law that models three rods of afreely jointed chain. �a� Histogram of the probability density of the spring length from a BD simulation

compared with the exact distribution. The time steps used were �t=10−5 �� and �t=10−6 ��. �b� The relativeerror in the second moment �r2�eq �� and the fourth moment �r4�eq �� from the BD simulations.

as approximating the inverse Langevin function by a simple rational function.


Our task for the remainder of this section is to develop a spring force law whichapproximates the RWS model, because it is that spring force law that exactly models thefreely jointed chain. We start by considering the equal rod case. We have shown that theRWS model can be used directly when each spring represents two or three rods. Wheneach spring represents four rods, the RWS model only has two regions of extension, witha simple form in each region, and so could also be easily implemented in BD simulations.

For the cases where each spring represents five or six rods the spring force laws havethree regions in extension. The force law in the third region �near full extension� is givenby Eq. �30�. We can develop a single functional form that can approximate the first tworegions. In this way we can approximate the RWS model which has three regions inextension by a spring force law which only has two regions and so is easier to implementin simulations. An approximate force law for the five-rod case is

fsA

kBT�

539r/225

1 − 3r2/5, 0 r 3/5, �42�

fsA

kBT=

1 + 2r

5r�1 − r�, 3/5 r 1. �43�

The parameters for the force law in the first region were chosen by first assuming afunctional form in r with two adjustable constants. The parameters were constrained sothat the force law is continuous at r=3/5 because the RWS model for five rods iscontinuous. The other free parameter was chosen to give a small error in the second andfourth moment of the spring length at equilibrium. The error in �r2�eq by using thisapproximate force law to model five rods is 0.02%. The error in �r4�eq is 0.0006%.

A similar method can be used to approximate the six-rod case. The approximate forcelaw is

fsA

kBT�

63r/25 − 49r3/125

1 − 9263r2/13500, 0 r 2/3, �44�

fsA

kBT=

1 + 3r

6r�1 − r�, 2/3 r 1. �45�

This approximate force law for the six-rod system has an error of 0.01% in �r2�eq and anerror of 0.02% in �r4�eq.

The advantage of still using two regions for the five- and six-rod cases is that we canutilize exactly the RWS model in the final region of extension. This allows us to build anapproximation that accurately represents many properties such as the second and fourthmoments of the spring length and also the response under large stretching.

For even larger number of rods per spring the final region near full extension whichhas a simple force law is only valid over a smaller and smaller region of extension. In thiscase the advantage to using two spring force laws spliced together does not seem greatand so we have chosen to develop a single approximate force law over the entire exten-sion range. While many possible forms are possible to approximate the true spring force
laws, we choose the form


fsA

kBT=

Cr + Dr3

1 − r2 , �46�

where A is the rod length and C and D do not depend on the fractional extension, r, butwill depend on the number of rods represented by the spring, �. The dependence on rtakes the same form as the Cohen force law �Cohen �1991��. We will see that with anappropriate choice of C and D, this can give an accurate representation of the freelyjointed chain. Because we have chosen the same form in r as the Cohen force law,calculations such as BD simulations can be easily and quickly modified to use the newforce law with no loss of computation speed but a uniformly more accurate result.

Although we are not using the RWS force law near full extension, we will still requirethat the limiting behavior of our approximate form match the limiting behavior of theRWS force law. The result of this choice will be that the approximate spring will behavelike the freely jointed chain in the limit of strong stretching. As the fractional extensionapproaches 1, the RWS force law diverges as

fsA

kBT→

1 − 2/�

1 − r. �47�

For our approximate force law to diverge in the same manner, we obtain the constraint onC and D of

C + D = 2 − 4/� . �48�

We must now develop a method for determining the remaining free parameter. Onenatural method would be to examine the RWS force law at small extension, find thecoefficient to r in that expansion which will be a function of �, and match this coefficientwith C. Because the constraint on C+D is to capture the strong stretching limit, therationale behind this choice would likely be so that the spring correctly captures theequilibrium limit. However, with this choice the second moment of the spring length atequilibrium, �r2�eq, is not correct. This is because, even at equilibrium, the spring samplesthe nonlinear regions of the force law, and the shape of the approximate force law doesnot exactly capture the shape of the RWS model.

Alternatively, one could calculate numerically the function C�� such that the secondmoment of the spring length for the approximate force law matches exactly the freelyjointed chain �which is also the same as the RWS model�. Recall that the second momentof the spring length is a key property that is important to capture �Underhill and Doyle�2004��. Not only is it related to the size of the coil at equilibrium, but also it is relatedto the zero-shear viscosity. In the constant force ensemble, the slope of the averageextension versus force curve at small force is also proportional to the second moment ofthe spring length.

Instead of having to deal with a numerical function C��, it would be better to have asimple approximate formula for C�� that still gives a very small error in the secondmoment of the spring length. Another reason an approximate form for C�� is sufficientis that even if we could make the error in the second moment vanish, the shape of thespring force law will limit the ability to capture other properties like the fourth momentexactly.

Using the asymptotic expansion developed in Underhill and Doyle �2004�, the second
moment of the spring length for the force law �Eq. �46�� with the constraint in Eq. �48� is


�r2�eq �3

�C−

30

�2C3 +30�24 − 7C + 2C2�

�3C5 + O 1

�4� . �49�

We want to choose C�� such that this second moment is the same as the freely jointedchain. The freely jointed chain has �r2�eq=1/�. By choosing

C = 3 −10

3�+

10

27�2 , �50�

the second moment of the spring length for the approximate spring force law �Eq. �46��is

�r2�eq �1

�+ O 1

�4� . �51�

The resulting approximate spring force law is

fsA

kBT=

�3 − 103� + 10

27�2�r − �1 + 23� + 10

27�2�r3

1 − r2 . �52�

To quantify the accuracy of this approximate spring force law, we need to compare it’sproperties with those of the underlying freely jointed chain. The average �r2�eq of thisspring force law is compared in Fig. 11�a� to the freely jointed chain it is meant torepresent as a function of the number of rods per spring. From the asymptotic expansionof the second moment, Eq. �51�, we see that the relative error should go like �−3 whichwe see explicitly from Fig. 11�a�. The approximate spring force law has an error of 0.01%at �=6 and smaller error for larger �. For reference we show the error if the Cohen forcelaw were used. An asymptotic expansion of the second moment of the spring length usingthe Cohen force law shows the relative error decays like �−1 as seen in the figure. Theslight changes made to the Cohen form have had a significant impact. The error has beenreduced by over a factor of 1000. The new force law can even be used down to four rodsper spring with only a small error.

The fourth moment of the spring length is also important to capture correctly becauseof it’s impact on zero-shear rheology. Similar to the second moment, we show in Fig.

FIG. 11. Relative error in the equilibrium moments of the spring length between an approximate spring forcelaw compared with an equal rod freely jointed chain. The solid lines are for the Cohen force law, and the dashedlines are for the new force law �Eq. �52��. �a� Relative error in the second moment, �r2�eq. �b� Relative error inthe fourth moment, �r4�eq.

11�b� the error in the fourth moment of the spring length for the approximate force law


with the error of the Cohen form for comparison. The fourth moment of the spring lengthfor the freely jointed chain is �r4�eq= �5�−2� / �3�3�. Again we see that the error of theapproximate force law �Eq. �52�� is very small, only about 1% at �=6 and smaller atlarger �. The error has been reduced by about a factor of 40 below that of the Cohen forcelaw.

The final property we analyze here is the force-extension behavior in the constantforce ensemble. Recall that the response of the freely jointed chain in this ensemble is the

Langevin function, �z�=L� f�. We compare the error of the approximate force law and theCohen force law in Fig. 12 for �=4,6 ,20,�. Note that the error does not exactly vanishin the limit �→� because the Cohen force law is not exactly the inverse Langevinfunction, although the error in fractional extension is less than 2% in that limit.

We have now developed and analyzed the error from using an approximate springforce law �Eq. �52�� to represent freely jointed chains. The error from using this approxi-mate force law instead of the RWS model is small enough that the dominant error in aBrownian dynamics simulations will likely be from a nonzero integration time step andstatistical error from a finite number of samples. The error is small even when eachspring represents only four rods. It is important to emphasize that the new force law �Eq.�52�� outperforms the Cohen force law at all levels of discretization and so should beused instead of the Cohen force law by future simulators.

We can develop a similar approximate force law that models freely jointed chains withunequal rod lengths by proceeding in the exact same manner. As with the equal rod case,we will assume a form like the Cohen force law

fsA

kBT=

Br + Gr3

1 − r2 . �53�

The values of B and G will depend on the number of rods and also the distribution of rod

FIG. 12. Relative error in the average fractional extension between an approximate spring force law comparedwith an equal rod freely jointed chain, where the number of rods each spring represents is �=4,6 ,20,� and thearrows denote increasing �. The dotted lines are for the Cohen force law, the dashed lines are for the newapproximate force law �Eq. �52��, and the solid line is the �=� limit of both force laws. The force is nondi-

mesionalized using the rod length, f = fA /kBT.

lengths. For the spring to have the correct limiting behavior at full extension we need


B + G = 2 − 4/� , �54�

where here � is defined using the average rod length, �=� / A, which is equal to the totalnumber of rods that the spring represents. For the equal rod case, we used the expansionof the second moment of the spring length for the approximate spring to get the remain-ing parameter B. Because the spring force takes the same form as the equal rod case, theexpansion of the second moment of the spring length is the same

�r2�eq �3

�B−

30

�2B3 +30�24 − 7B + 2B2�

�3B5 + O 1

�4� . �55�

We choose B in order to match this second moment with the second moment of the freelyjointed chain. The second moment of the freely-jointed chain is �r2�eq=A2 /�, where A2 is

a function of the distribution of rod lengths and is defined as A2�A2 / �A�2. Similarly we

will define A4�A4 / �A�4, which we will use later in discussing the fourth moment. Bychoosing

B =3

A2−

10A2

3�+

10A2�4A22 − 21A2 + 18�27�2 , �56�

the second moment of the spring length for the approximate spring force law �Eq. �53��is

�r2�eq �A2

�+ O 1

�4� . �57�

The resulting approximate spring force law is

fsA

kBT= � 3

A2−

10A2

3�+

10A2�4A22 − 21A2 + 18�27�2 r +

�2 − 4/��r3

1 − r2 . �58�

In the limit �→� this becomes

fsA

kBT=

3

A2r +

2r3

1 − r2 . �59�

We can quantify the accuracy of the approximate force-law by comparing the second andfourth moments of the spring length with the second and fourth moments for the under-lying freely jointed chain. The fourth moment of the freely jointed chain is �r4�eq

= �5A22�−2A4� / �3�3�. Even when expressing a unequal rod system in terms of a few

averages over the rods, the parameter space is large. We will quantify the error of theapproximate spring force law for the same system shown in Fig. 5. This system consistsof equal numbers of rods of length A and rods of length 5A, resulting in A2=1.444 andA4=3.862. With these parameters fixed, we change the number of rods, and examine theerror in the moments of the spring length in Fig. 13.

IV. SUMMARY AND OUTLOOK

We have examined a new method to generate coarse-grained models of polymers asbead-spring chains using the constant extension ensemble behavior of the polymer. Weapplied it to a number of toy problems to illustrate the mechanics of the method andimportant aspects of coarse-graining. Applying this method to the FJC polymer showed
why current bead-spring chains are incapable of modeling polymers at high discretiza-


tion. The analysis was applied to freely jointed chains in which the rods can have differ-ent lengths showing that as the spread in rod lengths increases, the response of the chaincan change dramatically. This change cannot be accounted for solely by using an effec-tive Kuhn length. We applied the method to construct a dumbbell model of actin fila-ments. This illustrated that the method in dumbbell form is applicable to the WLC as wellas showing it can be used directly from experimentally data. These new force laws havea significant impact on not only the force-extension behavior but also the rheologicalbehavior.

At the length and time scales that polymer kinetic theory aims to capture, the twocommon approaches are to use more detailed models such as the freely jointed chain orto use bead-spring chains where each bead represents a large segment of polymer. Be-cause the current bead-spring models, such as the Cohen force law or Marko-Siggia forcelaw, produce errors if they are pushed to high discretization, there exists a gap wherethere is no accurate coarse-grained version of the more detailed model. One method toreduce the size of this gap is to use an effective Kuhn length or persistence length. Theuse of this method has been examined in Underhill and Doyle �2004�. This method hassignificant limitations including that there is not a unique, “correct” choice of the effec-tive Kuhn or persistence length valid for all situations.

The advance of the PET method is that it bridges this gap in a systematic manner,showing how to calculate the force law at any level of coarse-graining. However, inpractice the force laws calculated from the PET method may be difficult to implement. Inthese situations it may be sufficient to use an approximate force law. The approximateforce law can have a simple functional form that is easy to use in calculations andcaptures the underlying model to high accuracy. We have shown examples of developingapproximate force laws, including Eq. �52� which is capable of modeling a freely jointedchain to high discretization �springs representing as few as four Kuhn steps�. Simulationsthat have used the Cohen force law can be easily modified to use this new force law, Eq.�52�, including the semi-implicit predictor-corrector method developed by Somasi et al.�2002�. Because the new force law is always better than the Cohen force law, Eq. �52�should always be used instead of the Cohen force law for modeling of freely jointedchains.

FIG. 13. Relative error in the equilibrium moments of the spring length between an approximate spring forcelaw compared with an unequal rod freely jointed chain. The solid lines are for the Eq. �59� force law, and thedashed lines are for the new force law �Eq. �58��. �a� Relative error in the second moment, �r2�eq. �b� Relativeerror in the fourth moment, �r4�eq.

Similar to that development of an approximate force law as an alternative to the RWS


model, it appears useful to approximate the coupling behavior of worm-like chains usingbending potentials. We hope that bead-spring chain models with bending potentials willbe used in the future to fill the current gap for worm-like chains.

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation CAREER ProgramGrant No. CTS-0239012 P.T.U. acknowledges support from the National Science Foun-dation Graduate Research Fellowship program.

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Documents

Development of bead-spring polymer models using …web.mit.edu/doylegroup/pubs/UnderhillJoR2005.pdfDevelopment of bead-spring polymer models using the constant extension ensemble Patrick